Having these inclusions, it is natural to ask when equality holds. Aalipour
et al. [1]} characterised groups whose enhanced power graph and
commuting graph are equal: these are precisely the groups having no subgroup
\(C_p\times C_p\)  for any prime \(p\)  (so the Sylow subgroups are cyclic or
generalized quaternion). Our next job is to refine this.
